Consider a set of n trials involving independent and identically distributed random variables X1;X2; : : : ;Xn; where the outcome of 1 corresponds to success and the outcome of 0 corresponds to failure, and the probability of success on each trial is p. (a) Show that E(Xi) = p, and Var(Xi) = p(1 p) for i = 1; 2; : : : ; n. (b) Let Y = X1 +X2 +: : :+Xn: Show that E(Y ) = np, and Var(Y ) = np(1p). (c) Let ^p = Y=n. Show that E(^p) = p, and Var(^p) = p(1 p)=n. Use facts from lecture notes and/or the Lohr Appendix handout to justify your reasoning for these problems.